OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
LINKS
EXAMPLE
The a(1) = 1 through a(6) = 11 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(31) (32) (33)
(112) (41) (42)
(211) (113) (51)
(212) (114)
(311) (213)
(312)
(411)
(2112)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@Differences[#]&]], {n, 0, 15}]
PROG
(PARI) \\ Row sums of R(n) give A179269 (breakdown by width)
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L, v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M, n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j, w]))); x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 27 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 10 2019
EXTENSIONS
a(26)-a(42) from Lars Blomberg, May 30 2019
Terms a(43) and beyond from Andrew Howroyd, Aug 27 2019
STATUS
approved